Geometric Analysis (course 742)

University of Maryland

Department of Mathematics

Autumn 2013

Time: Tuesdays, Thursdays at 11am.
Room: 1308 Mathematics Building.

Teacher: Y.A. Rubinstein. Office hours: Tuesdays, Thursdays 1:10-2pm and by appointment.

Course plan:
The goal will be to give an introduction to Geometric Analysis that is accessible to beginning students interested in PDE/Analysis or Geometry but not necessarily in both nor necessarily with background in both. Topics will range, e.g., from Calculus of Variations, Bochner technique, Morse theory, weak solutions and elliptic regularity, maximum principle for elliptic and parabolic equations, Green's function of the Laplacian, isoperimetric and Sobolev inequalities, continuity method, curvature and comparison results, harmonic maps, curvature prescription problems.

Requirement: each student taking the course for a grade will be asked to prepare and typeset notes for a block of lectures as well as the solutions of the homework exercises assigned during those lectures.

Main references:

T. Aubin, Some nonlinear problems in Riemannian geometry, Springer, 1998.
D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2001.
P. Li, Geometric Analysis, Cambridge University Press, 2012.
P. Petersen, Riemannian Geometry, Springer, 2006.
M.M. Postnikov, Geometry VI: Riemannian Geometry, Springer, 2001.
R. Schoen, S.-T. Yau, Lectures on Differential Geometry, Int. Press, 1994.
M. Struwe, Variational Methods, 4th Ed., Springer, 2008.

Additional references:

Ambrosio, Gigli, A user's guide to optimal transportation (available online).
Frederic Robert, Notes on the construction of Green's function on a Riemannian manifold.

Lecture notes:

Lectures 1-4 (Ryan Hunter)
Lectures 5-6 (Jacky Chong)
Lectures 9-10 (Jason Suagee)
Lectures 11-12 (Siming He)
Lectures 13-15 (Zhenfu Wang)
Lectures 16 & 19 (Siming He)
Lectures 17-18 (Bo Tian)
Lectures 20-21 (Bo Tian)


  • Lecture 1
    Overview. Basic definitions of Riemannian geometry. Langrangians and Euler-Lagrange equations. The length Lagrangian and its EL equation.

  • Lecture 2
    More basic definitions of Riemannian geometry. Parallel translation and the geodesic equation. Comparison with the E-L equation from last time.

  • Lecture 3
    Jacobi theory I.

  • Lecture 4
    Jacobi theory II.

  • Lecture 5
    The direct method in the calculus of variations. Compactness of sublevel sets as motivation for requiring weak sequential lower semicontinuity and coercivity.

  • Lecture 6
    Situations where the direct method can be applied: p-Laplacian, harmonic maps into Euclidean space (generalizing geodesics), optimal transportation plans.

  • Lecture 7
    Rockafellar's theorem. Basic convex analysis. The fundamental theorem of optimal transportation.

  • Lecture 8
    The fundamental theorem of optimal transportation - continued. Dual formulation. Brenier's theorem.

  • Lecture 9
    The Weitzenbock formula. Bochner's method for Killing fields and harmonic 1-forms.

  • Lecture 10
    Weak solutions and local elliptic regularity (L. Simon's lecture 6)

  • Lecture 11
    Interior Schauder estimates (L. Simon's lecture 12).

  • Lecture 12
    Interior Schauder estimates - continued.

  • Lecture 13
    Interior Schauder estimates - continued.

  • Lecture 14
    Maximum principle.

  • Lecture 15
    Maximum principle and Green's function in Euclidean space (and domains).

  • Lecture 16
    Green's function on a Riemannian manifold (F. Robert's notes).

  • Lecture 17
    Heat equation and kernel (L. Simon's lecture 11).

  • Lecture 18
    Heat equation and kernel - continued. Weyl's law.

  • Lecture 19
    Green's function on a Riemannian manifold - parametrix construction continued.

  • Lecture 20
    Nash, Moser, De Giorgi theory (L. Simon's lecture 17

  • Lecture 21
    Nash, Moser, De Giorgi theory - continued.

  • Lecture 22
    Applications of Nash, Moser, De Giorgi theory.

  • Lecture 23
    Applications of Nash, Moser, De Giorgi theory to nonlinear equations.